To prove Theorem REF  we follow closely Bary-Soroker–Fehm–Petersen's [1]} proof
for varieties.
However, we will have to prove a slight improvement of their fibration theorem (see Theorem REF  below) to deal with the more general situation of products of arithmetic schemes.
Bary-Soroker–Fehm–Petersen's product theorem for the Hilbert property over \(K\)  is a direct consequence of their following more general fibration theorem.
